Download A Mathematical Theory of Communication by Claude E. Shannon PDF

By Claude E. Shannon

Scientific wisdom grows at a stupendous pace--but few books have had as lasting an influence or performed as very important a job in our smooth international as The Mathematical conception of Communication, released initially as a paper on conversation concept within the Bell process Technical Journal greater than fifty years in the past. Republished in publication shape almost immediately thereafter, it has when you consider that passed through 4 hardcover and 16 paperback printings. it's a progressive paintings, remarkable in its foresight and contemporaneity. The collage of Illinois Press is happy and commemorated to factor this commemorative reprinting of a classic.

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35 2. , px y = px py (apart possibly from a set of points of probability zero). 3. Consider a generalized averaging operation of the following type: Z p0 y = with Z Z ; ax y dx = ; ax y px dx ; ax y dy = 1 ; ; ax y  0 : Then the entropy of the averaged distribution p0 y is equal to or greater than that of the original distribution px. 4. We have ; H x y = H x + Hx y = H y + Hy x and : Hx y  H y 5. Let px be a one-dimensional distribution. The form of px giving a maximum entropy subject to the condition that the standard deviation of x be fixed at is Gaussian.

0. Since the average signal power P is less than or equal S N C  W log 1 + P N  W log 1+ S N : S N and are limited to band W and peak S the result will be proved. Consider the p p ensemble of functions of the following type. A series of t samples have the same value, either + S or , S, then the p next t samplesphave the same value, etc. The value for a series is chosen at random, probability 12 for + S and 12 for , S. ), the output is peak limited to S. Furthermore the average power is nearly S and can be made to approach this by taking t sufficiently large.

The set of all functions of time containing no frequencies over W cycles per second. 3. The set of all functions limited in band to W and in amplitude to A. 4. The set of all English speech signals as functions of time. 1 For example with the set, f t  = sint + ; we may give a probability distribution for , P . The set then becomes an ensemble. Some further examples of ensembles of functions are: ; ;:::; n) with the probability of fk being pk . 1. A finite set of functions fk t  (k = 1 2 2.

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